Gauge symmetry and constraints structure for topologically massive AdS gravity: a symplectic viewpoint

Abstract

By applying the Faddeev-Jackiw symplectic approach we systematically show that both the local gauge symmetry and the constraint structure of topologically massive gravity with a cosmological constant , elegantly encoded in the zero-modes of the symplectic matrix, can be identified. Thereafter, via a suitable partial gauge-fixing procedure, the time gauge, we calculate the quantization bracket structure (generalized Faddeev-Jackiw brackets) for the dynamic variables and confirm that the number of physical degrees of freedom is one. This approach provides an alternative to explore the dynamical content of massive gravity models.

Symplectic realization of two interacting spin-two fields in three dimensions

“…There exists, however, a symplectic realization that provides straightforward effective tools for the study of the dynamical structure of Lagrangian field theories, which is geometrically well motivated and is based on the symplectic structure of the space-phase [55][56][57][58][59][60][61], and therefore it is different from Dirac's approach. The advantage of this construction is that we do not need to catalogue the constraints as first-and second-class ones.…”

“…Furthermore, by introducing an appropriate gauge-fixing procedure, we can obtain the structure of quantization brackets between the dynamical variable, the functional measure for determining the quantum transition amplitude and the number of physical DoF. In the light of these results, this symplectic method can provides powerful tools to deal with massive gravity and bi-gravity theories, where secondary, tertiary, or higher-order constraints are present, because it is algebraically simpler than the Dirac algorith [60,61].…”

“…which is first-order in time derivativeξ i , where hereafter ξ i stands for the collection of all the dynamical fields of the theory. The functionals ξ i are usually referred as symplectic variables in the symplectic framework [55][56][57][58][59][60][61]. Furthermore, a i [ξ] are the components of the so-called canonical one-form a = a i (ξ)dξ i as defined in ref.…”

“…In this paper we have discussed the construction of a symplectic realization for a theory describing two interacting spin-two fields in three dimensions, called Zwei-Dreibein gravity. The construction was done within the symplectic framework developed originally by Faddeev and Jackiw [55,60,61]. A central point in our discussion is that the analysis be focused on the properties of the pre-symplectic two-form matrix and its corresponding zero-modes which are associated with the constrained dynamics of the theory.…”

Symplectic realization of two interacting spin-two fields in three dimensions

“…There exists, however, a symplectic realization that provides straightforward effective tools for the study of the dynamical structure of Lagrangian field theories, which is geometrically well motivated and is based on the symplectic structure of the space-phase [53][54][55][56][57][58][59], and therefore it is different from Dirac's approach. The advantage of this construction is that we do not need to catalogue the constraints as first-and second-class ones.…”